(Extended) Cox model: \[\begin{align} \lambda_i(t)=\lambda_0(t) \exp(\beta Y_i(t) + \alpha Z_i + \eta W_i) \end{align}\]

where:

• \(\lambda_i(t)\) is the hazard function at time \(t\) for individual \(i\);
• \(\lambda_0(t)\) is the unspecified baseline hazard function;
• \(\alpha\) measures the effect of \(Z_i\) on the hazard function;
• \(\beta\) measures the association between the trajectory function \(Y_i(t)\) and the hazard function.

1. are measured at determined time points (\(t_{ij}\))
2. can have missing values over time
   • => Imputation?
   • \(\Rightarrow\) Bias introduction
3. are measured with some degree of error
   • \(\Rightarrow\) Noise in the biomarker trajectory (\(Y_i(t_{ij}) \neq X_i(t_{ij})\))
4. can be endogenous
   • \(\Rightarrow\) Trajectory can change when the event occurs
   • \(\Rightarrow\) Bias introduction

(Generalised) linear mixed effect (LME) model: \[Y_{i}(t_{ij})=X_{i}(t_{ij})+\epsilon_{i}(t_{ij})\]

where:

• \(Y_{i}(t_{ij})\) is the observed value
• \(X_{i}(t_{ij})\) is the true (unobserved) value of the longitudinal measurement at time \(t_{ij}\) for individual \(i\).
• \(\epsilon_{i}(t_{ij})\) is a random error term, usually:
  \[\epsilon_{i}(t_{ij})\sim \mathcal{N}(0,\sigma^2)\]

Test simultaneously an effect on:

• a biomarker (\(\gamma\));
• an event (\(\alpha\));
• a biomarker and an event (\(\beta\gamma+\alpha\)).

A gain in statistical power to detect those effects

• if \(\beta\neq0\) (to detect a joint effect of \(Z\): \(\beta\gamma+\alpha\neq0\)) Chen, Ibrahim, & Chu (2011);
• compared to the extended Cox model.

The standard (joint likelihood) formulation involves two components:

• a longitudinal component
• a time-to-event (survival) component.

Let:

• \(n\), the sample size;
  
• \(i\), an individual (\(i=1,\cdots,n\));
• \(m_i\), the number of measurements on individual \(i\);
• \(t_{ij}\), a time points (\(j=1,\cdots,m_i\)).

(Generalised) linear mixed effect (LME) model: \[Y_{i}(t_{ij})=X_{i}(t_{ij})+\epsilon_{i}(t_{ij})\] \(X_{ij}\) is the trajectory function, and could be defined: \[\begin{gather}X_{i}(t_{ij})=\theta_{0i} + \theta_{1i}t_{ij} + \cdots + \theta_{pi}t_{ij}^p &, & \boldsymbol\theta_p \sim \mathcal{N}(\boldsymbol\mu_, \boldsymbol\Sigma)\end{gather}\]

For simplicity here, we assume linearity over time (\(\theta_{0i}+\theta_{1i}t_{ij}\)): \[\begin{gather}Y_{i}(t_{ij})=\theta_{0i}+\theta_{1i}t_{ij}+\gamma Z_i+\delta W_i+\epsilon_{ij} &, & \boldsymbol{\theta} \sim \mathcal{N}_2 (\boldsymbol{\mu},\boldsymbol{\Sigma})\end{gather}\]

(Generalised) linear mixed effect (LME) model: \[Y_{i}(t_{ij})=\theta_{0i}+\theta_{1i}t_{ij}+\gamma Z_i+\delta W_i+\epsilon_{ij}\]

where:

• \(Y_{i}(t_{ij})\), the observed value;
• \(X_{i}(t_{ij})\), the true (unobserved) value of the longitudinal measurement at time \(t_{ij}\) for individual \(i\);
• \(\epsilon_{ij}\) is a random error term, usually: \(\epsilon_{ij}\sim \mathcal{N}(0,\sigma^2)\);
• \(Z_i\), a vector denoting the genotype of individual \(i\);
• \(W_i\), a set of adjusting covariates.

(Extended) Cox model (proportional hazards): \[\begin{align} \lambda_i(t)&=\lim_{dt \to 0} \frac{P\{t\leq T_i<t+dt|T_i\geq t, \bar{Y_i}(t), Z_i, W_i\}}{dt}\\ &=\lambda_0(t) \exp\{\beta X_{i}(t) + \alpha Z_i + \eta W_i\} \end{align}\]

Where:

• \(\bar{Y_i}(t)=\{Y_i(u),0 \leq u \leq t\}\), the history of the trajectory;
• \(T_i\), the event time for individual \(i\);
• \(C_i\), the right censoring time (end of the follow-up);
• \(\Delta_i\), an event indicator: \(\begin{cases} \Delta_i=0, & \text{if }\ T_i>C_i. \\ \Delta_i=1, & \text{if }\ T_i <= C_i. \end{cases}\)

Null hypothesis: \(\begin{cases}H_0&:& \theta=\theta_0\\H_1&:& \theta\neq\theta_0\end{cases}\)

• Likelihood Ratio Test
  \(LRT=-2\{\ell(\hat{\theta}_0)-\ell(\hat{\theta})\}\)

• Wald Test
  \(W=(\hat{\theta}-\theta_0)^\top \mathcal{I}(\hat{\theta})(\hat{\theta}-\theta_0)\)
  (univariate: \((\hat{\theta}_j-\theta_{0j})/\widehat{s.e.}(\hat{\theta}_j)\))

• Score Test
  \(U=S^\top(\hat{\theta}_0)\{\mathcal{I}(\hat{\theta}_0)\}^{-1}S(\hat{\theta}_0)\)

• Is Joint Model estimators good?
  Bias, variance and RMSE (Root-Mean Square Error)
  \[\begin{align} \operatorname{MSE}(\hat\phi)&= \operatorname{Bias}(\hat\phi)^2 + \operatorname{Var}(\hat\phi)\\ \operatorname{RMSE}(\hat{\phi})&=\sqrt{\operatorname{MSE}(\hat\phi)}\\ &=\sqrt{E\{(\hat{\phi}-\phi)^2\}} \end{align}\] \[\phi=(\beta, \gamma, \alpha)\]

• Can we do a whole genome analysis…
  … in a reasonable time frame?

What if, we split the job in two?

\(\Rightarrow\) "Two-Step" (Tsiatis, DeGruttola, & Wulfsohn, 1995)?

1. (Generalised) linear mixed effect (LME) model
   \[\begin{align} Y_{i}(t)&=X_i(t)+ \epsilon_{i}(t)\\ X^*_{i}(t)&=E\{X_{i}(t)|\bar{Y_i}(t), T_i\geq t\} \end{align}\]

2. (Extended) Cox model (proportional hazards)
   \[h_i(t)=h_0(t) \exp\{\beta X^*_{i}(t)\}\]

Let's keep it simple, i.e., without covariates:

• the trajectory: \(Y_{i}(t)=\theta_{0i} + \theta_{1i}t + \gamma Z_i + \epsilon_{i}(t)\)

• the event: \(\lambda_i(t)=\lambda_0(t) \exp\{\beta X_{i}(t) + \alpha Z_i\}\)

• the time of event, following the (classical) exponential distribution (Austin, 2012):

  \[\begin{gather} H_i(T_i)=\int_0^{T_i}\lambda_0(t) \exp(\beta X_i(t)+\alpha Z_i)dt , & \lambda_0(t)=\lambda\\ F_i(T_i)=1-exp(-H_i(T_i))=u , & u\sim\mathcal{U}(0, 1) \end{gather}\] \[T_i=\frac{1}{\beta\theta_{1i}}\log\left(1-\frac{\beta\theta_{1i}\times \log(1-u)}{\lambda \exp(\beta\theta_{0i}+(\beta\gamma+\alpha)Z_i)}\right)\]